nc FILE COPY
AD-A158 081
US ARMY
MATERIEL
COMMAND
TECHNICAL REPORT BRL-TR-2656
STEADY STATE PENETRATION OF RIGID
PERFECTLY PLASTIC TARGETS
Romesh C. Batra
Thomas W. Wright
May 1985
s
DTIC
ELECTE
JUL2 9 ©85
I
X B
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US ARMY BALLISTIC RESEARCH LABORATORY
ABERDEEN PROVING GROUND, MARYLAND
85 T 29 017
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REPORT DOCUMENTATION PAGE befoI^comple^form
I. REPORT NUMBER |2. GOVT ACCESSION NO. 3- RECIPI ENT'S CAT ALOG NUMBER
II. REPORT NUMBER
TECHNICAL REPORT BRL-TR-2656
4. TITLE («n< f Submit)
Steady State Penetration of Rigid
Perfectly Plastic Targets
S. TYPE OF REPORT ft PERIOD COVERED
6. PERFORMING ORG. REPORT NUMBER
17. AUTHORO)
Romesh C. Batra* and Thomas W. Wright
8. CONTRACT OR GRANT NUMBERf*)
9. PERFORMING ORGANIZATION NAME ANO ADDRESS
US Army Ballistic Research Laboratory
ATTN: AMXBR-TBD
Aberdeen Proving Ground, MD 21005-5066
II. CONTROLLING OFFICE NAME ANO ADDRESS
10. PROGRAM ELEMENT. PROJECT, TASK
AREA 8 WORK UNIT NUMBERS
IZ. REPORT DATE
US Army Ballistic Research Laboratory May 1985
ATTN: AMXBR-OD-ST is. number of pages
Aberdeen Proving Ground, MD 21005-5066 38
14. MONITORING AGENCY NAME 8 ADDRESS <7/ dllltltnt I root Controlling Olll cm) IS. SECURITY CLASS, (ot thlm rmport)
UNCLASSIFIED
is.. DECLASSI FI CATION /DOWN GRADING-
SCHEDULE
I 16. DISTRIBUTION STATEMENT (at thlm Rtpert)
Approved for public release; distribution unlimited.
[ 17. DISTRIBUTION STATEMENT (of th e mbetrmct entered in Block 20, II different from Rmport)
is. supplementary notes
University of Missouri - Rolla
Department of Engineering Mechanics
Rolla, MO 65401-0249 _
19. KEY WORDS (Contlnum on rmrmrem mldm it nmceeemry mnd Idmntlly by block number)
-30. ABSTRACT fCmatbeue mm reeeree ft necreemry mod Identify by block number)
-^The problem of steady penetration by a semi-infinite, rigid penetrator into an
infinite, rigid/perfectly plastic target has been studied. The rod is assumed
to be cylindrical, with a hemispherical nose, and the target is assumed to obc>
the Von Mises yield criterion with the associated flow rule. Contact between
target and penetrator has been assumed to be smooth and frictionless. Results
computed and presented graphically include the velocity field in the target,
the tangential velocity of target particles on the penetrator nose, normal
pressure over the penetrator nose, and the dependence of the axial resisting^
FORM
I JAM 71
EDITION OF I MOV 68 IS OBSOLETE
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE f*Nn D.I. Enttrtd)
f4, force on penetrator speed and target strengths
(
r o
TABLE OF CONTENTS
Pa
LIST OF ILLUSTRATIONS . ■
INTRODUCTION .
FORMULATION OF THE PROBLEM . I
FINITE ELEMENT FORMULATION OF THE PROBLEM . H
COMPUTATION AND DISCUSSION OF RESULTS . 1
CONCLUSIONS . 2
REFERENCES . 2
DISTRIBUTION LIST . 3
DTIC
SeLECTEn
JUL29 1985 j I
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DT,-: TAB
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.lability Code3
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! Special
LIST OF ILLUSTRATIONS
FIG. NO.
Page
1
2
3
i*
5
6
7
8
9
10
1 1
12
13
The Region to be Studied . 11
Finite Element Grid Used . 14
Velocity Field in the Target Material (a = 4.0) . 15
Normal Stress Distribution on the Hemispherical Nose
of the Penetrator . 16
Axial Force vs. a . 18
Contours of p in the Target . 19
Variation of -a at Points of the Target Along the Axis
zz
Ahead of the Penetrator . 20
Tangential Velocity Distribution on the Hemispherical
Nose of the Penetrator . 21
Variation of the z-Velocity of Target Particles Along the
Axis Ahead of the Penetrator . 22
Variation of v With r on the Surface z = 0 . 24
z
Variation of v With r on the Surface z = - 1.595. .... 25
z
Comparison of Stresses Calculated in this Paper and in
Pidsley7 . 26
Contributions to the Integral Formula (12) . 28
5
PREVIOUS PAGE
IS BLANK
I . INTRODUCTION
In simple theories of penetration the material properties of target
and penetrator are often represented only by constant characteristic
stresses, as for example in Tate.1*2 Although this approach leads to
results that are qualitatively correct, it can be difficult to use quanti¬
tatively. Some of the problems have to do with actual deformations in
target and penetrator including lateral motion, and others are associated
with the fact that the plastic flow stress is determined only by the devia-
toric components of stress whereas the spherical or pressure component,
which may be quite large ahead of the penetrator and contributes signifi¬
cantly to the retardation of the penetrator, is unrelated to flow stress.
These and other matters have been discussed recently in some detail by
Wright . 3 It would be desirable to account for lateral motion and hydro¬
static effects in some simple way, but at present the details are insuffi¬
ciently known to suggest high quality approximations that might be suita¬
ble. In developing an engineering model for penetration and perforation,
Ravid and Bodner4 have attempted to meet this difficulty be assuming simple
kinematics for the flow around the penetrator and then adjusting some
unknown parameters so as to minimize the plastic dissipation. They charac¬
terize this procedure as being "a modification of the upper bound theorem
of plasticity to include dynamic effects," but even if 3uch a modified
theorem is actually valid, at present there is no way to tell how close
such a bound might be.
I
--•In this paper a detailed numerical solution to an idealized penetra¬
tion problem is presented in an attempt to shed some light on these
matters^ The approach taken is as follows. Suppose that the penetrator is
in the intermediate stages of penetration so that the active target/
penetrator interface is at least one or two penetrator diameters away from
either target face, and the remaining penetrator is still much longer than
several diameters and is still traveling at a speed close to its striking
velocity .
This situation is idealized here in several ways. First, it is
assumed that the rod is semi -infinite in length and that the target is
infinite with a semi- infinite hole. Furthermore, it is assumed that the
rate of penetration and all flow fields are steady as seen from the nose of
the penetrator. These approximations are reasonable if the major features
lTate, A., "A Theory for the Deceleration of Long Rods After Impact," J. Mech.
Phys. Sol. 15, 1967, 387-399.
zTate, A., "Further Results in the Theory of Long Rod Penetration , " J, Mech.
Phys. Sol. 17, 1969, 141-150.
3 Wright , T.W. , "A Survey of Penetration Mechanics for Long Rods," in Lecture
Notes in Engineering , Vol. 3, Computational Aspects of Penetration Mechan¬
ics, Eds., J. Chandra and J. Flaherty, Spring er-Verlag , New York, 1983.
URavid, M. and Bodner, S.R. , "Dynamic Perforation of Visco-plastic Plates by
Rigid Projectiles," Int. J. Eng. Sci. 21, 1983, 577-591.
7
PREVIOUS PAGE
IS BLANK
of the plastic flow field become constant within a diameter or so of the
nose of the penetrator, and will be justified a posteriori by the calcula¬
tion.
Next, it is assumed that no shear stress can be transmitted across the
target/penetrator interface. This is justified on the grounds that a thin
layer of material at the interface either melts or is severely degraded by
adiabatic shear. Thi3 assumption, together with the previous one, makes it
possible to decompose the problem into two parts in which either a rigid
rod penetrates a deformable target or a deformable rod is upset at the
bottom of a hole in a rigid target. Of course, in the combined case the
contour of the hole is unknown, but if it can be chosen so that normal
stresses match in the two cases along the whole boundary between penetrator
and target, then the complete solution is known irrespective of the rela¬
tive motion at the boundary. Even without matching the normal stresses, it
would seem that valuable qualitative information about the flow field and
distribution of stresses can be gained if the chosen contour is reasonably
close to those that are actually observed in experiments.
Finally, the deforming material is assumed to be rigid/perfectly plas¬
tic. This assumption should be adequate for examining the flow and stress
fields near the penetrator nose, but will lose accuracy with increasing
distance, since it forces the effects of compressibility and wave propaga¬
tion to be ignored.
In this paper only the case of the deforming target and a rigid pene¬
trator is considered, where the penetrator is assumed to have a circular
cylindrical body and a hemispherical nose.
II. FORMULATION OF THE PROBLEM
With respect to a set of cylindrical coordinate axes fixed to the
center of the hemispherical nose of the rigid cylindrical penetrator,
equations governing the deformations of the target are
divy = 0,
diva - py = p(v-grad)v.
(1)
Here q is the Cauchy stress tensor, p is the mass density of the target
material, y is the velocity of a target particle relative to the pene¬
trator, which has absolute velocity vQ e, e being a unit vector along the
axis and in the direction of motion of the penetrator. The operators grad
and div signify the gradient and divergence operators on fields defined in
the present configuration. Equation (1)1 expresses the balance of mass
and implies that the target undergoes only volume preserving deformations
so that the mass density of the target stay3 constant. Equation (1)2
expresses the balance of linear momentum in the absence of body forces and
holds in all Galilean coordinate systems. In particular it holds in one
that translates at the constant velocity of the penetrator. Equation (1)^
8
holds only in such a translating system where all field variables are inde¬
pendent of time. The target material is assumed to obey the Von-Mises
yield criterion and the associated flow rule. That is (Prager and Hodge),5
0
i o _
5 = - pi + - D ,
Vn '
D = (gradv + (gradv)^)/2 (2)
I = \ tr (D2) .
In equations (2) p is the hydrostatic pressure (which, of course, cannot be
determined by the deformation because of the assumption of incompressibil¬
ity), 1 is the identity matrix, Q is the strain rate tensor, aQ is the
flow stress of the target material in simple compression and tr(P2) equals
the sum of the diagonal terms of the square matrix D. Equation (2)1 is the
constitutive relation of an incompressible Navier-Stokes fluid with viscos¬
ity coefficient equal to o / 2\/3I. Equations (2), when substituted into
(1)^, give the field equation
T
- grad p + aQdiv( (gradv + (gradv) )/2v/3T) = p(v-grad)v (3)
which together with (1)1 is to be solved for p and v subject to suitable
boundary conditions. Before stating these the following non-dimensional
variables will be introduced.
Q = <7/°0» Y = y/vQ, f = r/rQ, z = z/rQ, p = p/aQ .
The pair (r,z) denotes the cylindrical coordinates of a point with respect
to axes attached to the center of the hemispherical nose with the positive
z-axis pointing into the target material. Rewriting equations (1)1 and (3)
in terms of non-dimen3ional variables, dropping the superimposed bars, and
agreeing to denote the gradient and divergence operators in non-dimensional
coordinates by grad and div, we arrive at the following set of equations.
divv = 0
T (4)
- grad p + div ((gradv + (gradv) ) /2 VXD = <*(v grad)v,
2
where a = ppv /oo is a non-dimensional number.
5 Prager, W. and Hodge, P.t Theory of Perfectly Plastic Solids y Dover Pub l
New York, 193d.
9
Boundary conditions must be given both for the penetrator/target
interface and for points remote from the penetrator. As stated in the
introduction the interface conditions are
t • (on) = 0,
v • n = 0,
(5)
where n is a unit normal on the surface and t is a unit tangent on the
surface. At points far away from the penetrator.
2 2 1/2
jv + v e! ■+ 0 as I x I = (r + z ) -*■ ", z > -
i ~ o~ 1
| on | 0 as z + -
(6)
where e is a unit vector in the positive z-direction, as before. The
boundary conditions (5) state that the contact surfaces between target and
penetrator are smooth and there is no interpenetration of the target materi¬
al into the penetrator or vice versa. The boundary condition (6)^ is
equivalent to the statement that target particles at a large distance from
the penetrator appear to be moving at a uniform speed with respect to it.
Equation (6)^ states that far to the rear the traction field vanishes.
Note that the governing equations (3) are nonlinear in v and that a solu¬
tion of (1)1 and (3) under the stated boundary conditions, if there is one,
will depend on the rates at which the quantities in (6) tend towards zero.
III. FINITE ELEMENT FORMULATION OF THE PROBLEM
In order to solve the problem numerically, it is possible to consider
only a finite region of the target, and since deformations of the target
are axisymmetric, only the target region shown in Figure 1 is studied.
Whether the region considered is adequate or not can be easily decided by
solving the problem for two different values of the parameter a. If the
two solutions so obtained are essentially equal to each other in the vicin¬
ity of the penetrator, then the region studied is sufficient and the effect
of boundary conditions at the outer surface EFA has a negligible effect on
the deformations of the target material in close proximity to the penetra¬
tor. The boundary conditions imposed on the finite region are
a = 0, v =0 on the bottom surface AB,
zz r
t • an = 0, v ■ n = 0 on the common interface BCD,
(7)
a = 0, v =0 on the axis of symmetry DE,
T Z T*
v =0, = -1.0 on the bounding surface EFA.
A weak formulation of the problem is now obtained. Let$ be a smooth,
vector valued function defined on the region R of the target shown in
Figure 1, where $ satisfies the velocity boundary conditions included in
equations (7)1 - (7)^ and 6 = 0 on the surface EFA. In addition let 'P be
a bounded, scalar valued function defined on R. Taking the inner product
of both sides of equation ( H 1 with and of equation (4),, with <p, inte¬
grating the resulting equations over P, using the divergence theorem, the
stress boundary conditions in (7) and the stated boundary conditions for 4> ,
we arrive at the following equations.
J^P (divv) dV = 0 ,
f p (div<t> ) dV + f — - D : (grad<J> + (gradi)T) dV
« ' Jr2/T *
= a I (v ■ grad) v - 4> dV .
"he boundary value problem defined by equations (4) and (7) is equiva¬
lent. to the statement that equations (8) hold for every $ and ip such that
grad <t> and <4 are square integrable over R, d> satisfies the stated homogene¬
ous boundary conditions, and v satisfies all the velocity boundary condi¬
tions stated in (7).
An approximate solution of equations (8) has been obtained by using
the finite element metnod (see Becker, Carey, and Oden6 for details).
Oince equation (8)^ is nonlinear in v, the following iterative technique
has been used.
JRv(divvm) <
pm(div?) dV +
J h
[)m: (grad<£ + (grad^)^) dV
grad)v
■ckcr, Carey, G.,and Oden, *7.7..
: 1 TJrrl 7. if.
Finite Elements, An Introduction 3
-Hall, Enjlewooi Cl, iffe , //VT, 1981 .
RELATIVE Z-VEIOCITY
R-COORDINATE
Figure 11: Variation of With r on the Surface z = - 1.595.
25
deformation extends only to about one or two radii away from the nose,
whereas the stress is still significant at three radii. The nondimen-
sional values ofvT* computed for a = 6.15, become as large as 2.0 or 2.5
at points close to the nose tip. Since true strain rates scale with v /r ,
5-1 °
actual rates in the target may easily be of the order of 10 s or more for
reasonable values of v and r . Thus, strain rate effects may become
oo
important in some cases. This should be borne in mind especially for small
scale experimental studies, which will accentuate rate effects and tend to
make target materials appear stronger in small scale than in full scale.
Figures 10 and 11 show the variation of vz with r at z = 0 and
z = -1.595, respectively. These results indicate that more of the target
material at the sides of the penetrator deforms at higher values of a , even
though this is not true ahead of the penetrator as noted in the discussion
of Figure 9. In both Figure 10 and 11 the nondimensional velocity near the
penetrator decreases in absolute value with increasing a in order to sat¬
isfy the balance of mass. That is to say, since the deformations of the
target are volume preserving, the areas between each curve and v^ = -1.0
must be constant and just large enough to account for the rate of increase
of hole volume as the penetrator advances into the target. Thus the effect
of inertia is to spread the deformation farther to the sides, for which
compensation must be made closer in so a3 to maintain incompressibility.
In a recent paper, Pidsley7 described an unsteady calculation for one
impact velocity in which both target and penetrator were assumed to be com¬
pressible and elastic/perfectly plastic. He shows that after a few diame¬
ters penetration, the rate of penetration slows down and approaches a
steady 3tate. Figure 12 compares his values of pressure with the present
values of pressure and axial stress along the centerline ahead of the pene¬
trator for nearly the same values of a. Compressibility apparently has the
effect of reducing the pressure directly in front of the penetrator and
increasing it at distances greater than one penetrator radius or so, but
even so, the results seem to be broadly similar. ^It has not been possible
to compare velocity fields. In his paper Pidsley also notes that if the
equation of motion for steady flow is integrated along the central stream¬
line, there is a contribution from transverse gradients of shear stress,
unlike the case for a perfect fluid. This fact, which was also noted by
Wright,3 may be expressed in the following formula.
i 9 r
Ipv + p - szz - 2J ~^dz - - ozz(0) (12)
7~'idsley, p, H.t "A Numerical Study of Long Rod Impact Onto a Large Target , "
I. Mech. Phye. Sol. 32, 1984, 315-353.
RELATIVE Z-VELOCITY
o o-.
1 o 4
0 0 :
Figure 9
-1 - — I" - — — - 1 - r-
2 0 3.0 4.0 5 0
DISTANCE FROM THE NOSE TIP
Variation of the z-Velocity of Target Particles
Along the Axis Ahead of the Penetrator.
22
u-i - 1 - 1 i - r— - 1 - 1 - 1 —
0 10.0 20.0 30.0 40 0 50 0 60 0 70 0
ANGULAR POSITION
Figure 8: Tangential Velocity Distribution on the
Hemispherical Nose of the Penetrator.
80 0
21
* * ) /YIELD STRESS
VS. DISTANCE FROM THE NOSE
9°
8.0
7.0 i
5.0 j
DISTANCE FROM THE NOSE
Figure 7: Variation of -ozz at Points of the Target Along
the Axis Ahead of the Penetrator.
20
□ = 0.13
— r-
2.0
—i - r
4.0 6.0
R-AXIS
6
of p in the Target.
9
2 5
Figure 5
3.5
ALFA
Axial Force vs
18
Table 1 . Legend for Figures
Curve Type a
. 0.72
- . - 2.00
■ . 4.00
- 5.43
6.15
The total nondimensional force (average stress/flow stress) that acts
on the penetrator nose in the negative axial direction is given by
F =
an) sin20 d0.
Figure 5, which is a plot of F versus a, shows that F increases only
weakly and nearly linearly with a. A close approximation to the line is
given by the equation
F * 3.903 + 0.0773a , (11)
so that in typical impact problems, where the rate of penetration lies
roughly in the range 2 S a < 6, the retarding force varies only from about
4.1 to 4.4 times the product of cavity area and compressive flow stress in
the target material. Of course this range may change with nose shape, but
it doe3 seem to indicate why the choice of constant target resistance in
the simple theory of Tate1’* gives such good qualitative results. Note
also, that for the same range of a, the centerline stress on the penetrator
nose, as shown in Figure 4, varies from about 5.3 to 8.8 or as much as
twice the average value.
That a significant contribution to the axial force is made by the
spherical component of the Cauchy stress 2 is clear from Figure 6, which
shows values of non-dimensional p in the target. Whereas the deviatoric
components of a have magnitudes comparable to the flow stress oq, the
spherical component p is more than 8 times oQ near the nose tip. Figure 7
shows the principal stress component -c along the axis in front of the
z z
penetrator and demonstrates that stress falls rapidly with distance. The
stress near three radii for the smaller values of a cannot be accurately
calculated since the velocity gradient there is extremely small.
Figure 8 shows that the nondimensional velocity of target particles,
tangential to the penetrator nose, is essentially independent of a, and
Figure 9 shows that the same is true for the axial velocity of target
particles along r = 0 ahead of the penetrator. Note that the velocity
falls more rapidly than stress ahead of the penetrator so that target
17
NORMAL STRESS/YIELD STRESS
f
s
a
7 N
R
P
Figure 4: Normal Stress Distribution on the Hemispherical
Nose of the Penetrator.
16
.v.v-y->v-v
where m is the iteration number. For a < 2, the initial solution was taken
to be zero everywhere, and for a _ 2, the solution for a smaller value of a
was taken as the initial solution. The iterative process was stopped when,
at each nodal point,
||y" - y"'1 || < 0.01 Hy"-1 II. (10)
2 2 1 /2
where the norm is defined by ||v|| = (v* + v ) ' •
IV. COMPUTATION AND DISCUSSION OF RESULTS
A computer code employing 6-noded isoparametric triangular elements
has been written to solve the problem described above. Both the trial
solution (v,p) and the test functions (4,^) are taken to belong to the same
space of functions. Whereas, for the triangular element, v is defined in
terms of its values at all 6 nodal points, the pressure field p is defined
only in terms of its values at the corner nodes. The integrations-»An
equations (6) are performed by using the 4-point Gaussian quadrature rule.
Since the curved surface of the penetrator nose is not a natural coordinate
surface for the cylindrical geometry, it was found to be easiest to enforce
the boundary conditions there by using a Lagrange multiplier technique.
The accuracy of the developed code has been established by solving a
hypothetical flow problem for an incompressible Navier-Stokes fluid with
uniform viscosity. A body force field was calculated so as to satisfy the
balance of linear momentum exactly for an assumed, analytically known
velocity field, where the assumed velocities had the essential features of
those expected in the penetrator problem. Then the code wa3 used to com¬
pute the velocity and pressure fields for that body force. The computed
fields agreed very well with those known analytically. An important dif¬
ference between the test problem and the penetration problem is that in the
former the shear viscosity is taken to be constant, whereas in the latter,
it depends on the rate of deformation. Since only a simple modification in
the computer code is needed to incorporate this feature, it seems reason¬
able to assume that the computed solution is close to an analytical solu¬
tion of the problem.
Figure 3 shows the velocity field in the target material for a = 4.0.
The velocity fields for other values of a have a similar pattern. Target
points that lie to the rear of the center of the penetrator nose move
parallel to the axis of the penetrator. Target points that lie ahead of
the penetrator nose and within one penetrator diameter from it have a
noticeable radial component of velocity. The distribution of normal trac¬
tion on the penetrator nose for various values of a is plotted in Figure 4.
(See Table 1 for identification of the various lines in this and subsequent
figures.) Whereas the stress increases with a at the nose tip, it
decreases at the sides of the nose. The value of the normal stress for
0 = 45° seems to be independent of a , at least for the range of values of
a studied. For a = 6.15 the normal stress at approximately 6 = 83° becomes
negative, indicating a tendency for the target material to separate from
the penetrator nose. Since our formulation of the problem does not allow
for separation to occur, we seem to have reached the upper limit for the
validity of the calculation, at least for the hemispherical nose shape.
13
Each term is evaluated on r = 0, s is the deviatoric component of stress,
z z
and z is measured from the tip of the nose. Figure 13 shows the contribu¬
tions from the various components in this formula as computed for a = 5.43.
Since the target material becomes nearly rigid a short distance away from
the penetrator nose, the computation of the integrand in (12), which
requires differences and divisions with small numbers, is unreliable for
z > 0.6 or so, so that after that point, the upper bounding line was simply
extended horizontally. Note that the integral term in (12) contributes
substantially to the total and that the deviatoric component seems to stay
constant at approximately 0.75 out of a total of 8.5.
Since the target deformation is essentially zero at some distance
inside the boundary EFA, and since deformations are essentially independent
of z near the boundary AB, it seems reasonable to assume that the target
region chosen for computations is sufficient to obtain a good description
of the deformation in the vicinity of the penetrator nose.
V. CONCLUSIONS
For the range of values of a studied, noticeable deformation of the
target material occurs only at points that are less than three penetrator
radii away from the penetrator, and the target seems to deform farther to
the side than ahead of the penetrator. The target material adjacent to the
sides of the penetrator appears to extrude rearwards in a uniform block
that is separated from the bulk of the stationary target by a narrow region
with a sharp velocity gradient, but the highest strain rates occur just
ahead of the penetrator nose.
Maximum normal stresses occur at the nose tip, as might be expected,
and fall off rapidly away from that point. At the higher values of a, flow
separation seems to be indicated at the sides of the nose. The retarding
force was found to be a weak linear function of a, and gradients of shear
stress were found to make a strong contribution to the momentum integral
along the axial streamline.
S /
-The kinematics and stress fields found in this paper should prove use¬
ful in devising or checking the results from simpler engineering theories
of penetration^
27
REFERENCES
Tate, A., "A Theory for the Deceleration of Long Rods After Impact," J.
Mech. Phys. Sol. 15, 1967, 387-399.
Tate, A., "Further Results in the Theory of Long Rod Penetration," J.
Mech. Phys. Sol. 17, 1969, 141-150.
Wright, T. W., "A Survey of Penetration Mechanics for Long Rods," in
Lecture Notes in Engineering, Vol. 3, Computational Aspects of
Penetration Mechanics, Eds., J. Chandra and J. Flaherty, Springer-
Verlag, New York, 1983.
Ravid, M. and Bodner, S. R. , "Dynamic Perforation of Visco-plastic Plates
by Rigid Projectiles," Int. J. Eng. Sci. 21, 1983, 577-591.
Prager, W. and Hodge , P. , Theory of Perfectly Plastic Solids, Dover
Publ., New York, 1968.
Becker, E., Carey, c. , and Oden, J. T. , Finite Elements, An Introduction,
Vol. 1, Prentice-Hall, Englewood Cliffs, NY, 1981.
Pidsley, P. H., "A Numerical Study of Long Rod Impact Onto a Large
Target," J. Mech. Phys. Sol. 32, 1984, 315-333.
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